# Lurie’s theorem

$\newcommand{\Mell}{\mathcal{M}_\textbf{ell}}$ In the previous post, we showed that $\Mell\to \mathcal{M}_\textbf{fg}$ is a flat map, and stated the Hopkins-Miller theorem, namely that there exists a sheaf $\mathcal{O}^\textbf{top}$ of $\mathbf{E}_\infty$-rings on $\Mell$. $\newcommand{\co}{\mathcal{O}^\textbf{top}}$ The global sections of this sheaf is an $\mathbf{E}_\infty$-ring spectrum $TMF$.

$\newcommand{\dX}{\mathfrak{X}}$ As we’ll be using the term ‘‘derived stack’’ a lot, it’s probably a smart idea to present a definition.

$\newcommand{\Eoo}{\mathbf{E}_\infty}$ Definition: A derived stack $\dX$ is a Deligne-Mumford stack $X$ along with a sheaf $\co$ of $\Eoo$-rings on the etale site of $X$ such that $\pi_0 \co$ is the structure sheaf of $X$ (which, recall, is defined by the colimit of the structure sheaves of the affine etales over $X$). We’ll call $\dX$ a ‘‘derived lift’’ of $X$.

$\newcommand{\M}{\mathcal{M}}$ Lurie’s theorem provides a slick method of derived stacks. Let’s recall his theorem in the generality stated in the Behrens-Lawson book on topological automorphic forms.

Theorem (Lurie): $\newcommand{\m}{\mathfrak{m}}$ Let $(R,\m)$ be a local ring in mixed characteristic, $\DeclareMathOperator{\spf}{Spf}$ and let $\dX$ be a locally Noetherian separated Deligne-Mumford stack over $\spf R$. $\newcommand{\GG}{\mathbf{G}}$ Let $\GG\to \dX$ be a $p$-divisible group of height $n$ and dimension $1$, and let $\M_p(n)$ denote the moduli of $p$-divisible groups of dimension $1$ and height $n$. $\newcommand{\Eoo}{\mathbf{E}_\infty}$ If the associated map $\dX\to \M_p(n)$ is formally etale, or, equivalently, if the following condition is satisfied:

• for some etale cover $f:X\to \dX$, and for every $x\in X^\wedge_{\m}$, the map $X^\wedge_x \to \mathrm{Def}_{(f^\ast \GG)_x}$ classifying the deformation $(f^\ast \GG)|_{X^\wedge_x}$ of $(f^\ast\GG)_x$ is an isomorphism of formal schemes

then there is an even periodic presheaf $\co$ of $\Eoo$-rings on $\dX$ that is a sheaf in the etale topology that makes $(\dX,\co)$ a derived stack.

# Examples?

The reason this theorem was proved was to provide another, more canonical, in some sense, construction of the derived moduli stack of elliptic curves. Indeed, we get a sheaf of $\Eoo$-rings $\co$ on $\Mell$ (note that $(\Mell,\co)$ is the same as the lift of $\Mell$ constructed by Hopkins-Miller, because Lurie also asserts that the space of derived lifts of $\Mell$ is path-connected and has a preferred basepoint) from the following theorem of Serre-Tate.

Theorem (Serre-Tate): The deformation theory of an elliptic curve $E$ is controlled by the deformation theory of its corresponding $p$-divisible group $E[p^\infty] = \text{colim }E[p^n]$. In other words, the map $\Mell\to \M_p(n)$ is formally etale.

$\newcommand{\LT}{\mathrm{LT}}$ There is a more tautological example, which the theorem’s statement is clearly trying to emulate. In particular, we can consider the Lubin-Tate space, which parametrizes deformations of formal groups, in the sense that if $k = \mathbf{F}_{p^n}$ is a perfect field and $\GG$ a formal group of height $n$ over $k$, there is a bijection

$\newcommand{\Z}{\mathbf{Z}}$ By Lubin-Tate theory, we know that $\LT_n$ is a formal affine scheme over $\Z_p$, and there is a (noncanonical!) isomorphism

where the $u_i$ are defined by $[p]_{\widetilde{\GG}}(x) \equiv u_i x^{p^i}\bmod(p,u_1,\cdots,u_{i-1})$ – this is why they are noncanonical – where $\widetilde{\GG}$ denotes the universal deformation of $\GG$. The ring $W(k)[[u_1,\cdots,u_{n-1}]]$ is a complete regular local Noetherian ring, with maximal ideal $(p,u_1,\cdots,u_{n-1})$. That’s a regular sequence, and so in particular, there is a homotopy commutative even periodic Landweber exact ring spectrum $E$ (I’ll suppress the $n$ from the notation) such that $\pi_\ast E\simeq W(k)[[u_1,\cdots,u_{n-1}]][\beta^{\pm 1}]$, where $|\beta| = 2$. This is the famous Morava E-theory at the height $n$ (and the prime $p$).

When $n=1$, for instance, this coefficient ring is $\Z_p[\beta^{\pm 1}]$, and $E$ is $p$-adic complex $K$-theory. However – and this is confusing – the formal group laws are not the same. The formal group law associated to $p$-adic complex $K$-theory is the multiplicative group law $x+y+\beta xy$, while the formal group law associated to $E$ is the $p$-typification of the multiplicative formal group law.

Anyway, we get a derived lift of $\LT_n$ from the following theorem of Goerss-Hopkins-Miller.

Theorem (Goerss-Hopkins-Miller): The homotopy commutative ring spectrum $E$ is an $\Eoo$-ring spectrum.

$\newcommand{\frakm}{\mathfrak{m}}$ The structure sheaf is straightforward to construct: given an etale cover $U = \spf R \to \LT_n$, define $\co(U\to \LT_n)$ by using Lurie’s theorem that the affine etale site of $\pi_0 R$ is the same as the affine etale site of $R$. This derived stack $(\LT_n,\co)$ is denoted $\spf E$.